Cheong et al. [1]} have shown that every fan-crossing
free embedding of an \(n\) -vertex graph with \(4n-8\)  edges is 1-planar.
Such graphs are called extreme or optimal 1-planar
[2]}. The embedding consists of a 3-connected planar
subgraph, in which each face is a quadrangle with a pair of
crossing edges. Such graphs exist for \(n=8\)  and for all \(n \ge 10\) 
[2]}. They have a unique 1-planar embedding, except for
extended wheel graphs \(XW_{2k}\)  for \(k \ge 3\) , which have two
embeddings, where the poles are exchanged [4]}. An
extended wheel graph \(XW_{2k}\)  consists of two poles \(p\)  and \(q\)  and
a circle of length \(2k\) . There is an edge between each pole and each
vertex of the circle, whereas there is no edge \(p q\) . In
addition, there is an edge between a vertex of the circle and the
vertex after next (in cyclic order). Each of these edges is crossed
by an edge incident to a pole. Note that optimal 1-planar graphs can
be recognized in linear time [5]}.
